Comments on H. Volland's remarks on Austin's formula

Abstract

In a rece nt paper (publis hed in Nachr. T echn. Zeitschr.) Volland proposes an a nalyti cal model of the earth· ionos phe re waveguid e which is claimed to be in basi c agreem e nt with th e predictions of t he se mi e mpirica l transmi ssion formula of L W. Au stin for frequ e ncies less than 300 kc/s. In this note, th e Rat·earth as s umption s mad e by Volland are que stion e d an d any agreement of hi s res u lt with Austin's formula is a ttribut ed to a fortuitou s can cell ation of e rrors.It is rathe r interes tin g to n ote that H . Volland [1964] has r ece ntly publi shed a pa per und er th e title "Bemerkungen zur Aus tin'schen Forme]" (Re mark s on Austin' s formula). This basi c radio transmission formula was proposed b y L. W. Austin and pub]jshe d in the Bulle tin of th e Bure au of Standards over 50 ye ars ago [Aus tin , 1915]_ Volland proposes an analytical model of th e earth-ionosph ere waveguid e whi c h is claimed to be in basic agreem e nt with the predictions of the semi-e mpirical A us tin formula for frequ en cies less than 300 kc /s. At the sa me time , Volland sta tes that a fl at-earth ass umption is adequate and he asserts that the curvature of th e earth leads onl y to secondorder correc tion s and, thu s, ma y be neglected. Sin ce this latter state me nt is in violent co ntradicti on to mos t of the r ecent th eore ti cal work on VLF and LF mode theory of propagati on, so me comments on Volland's paper are called for.To pro perly orie nt the reader , a bri ef s urvey of Volland's formulation is give n. Hi s model consists of a flat homoge neo us earth with a di electri c constant Ee and conductivity (Te. The ionos phere is re prese nted by a reflecting le vel at a height h above the earth's surface_ His formula , in cylindrical coordinates 2khReRi aC (' = ('11 (2)The "eige nv alu e" CII of th e nth mode is to be de termin ed from(n= O, 1,2, . . . );CII can b e interpreted as the cosine of th e (complex) angle of in cidence 811 of th e nth mode in th e waveguid e. Now, it may be remarked that (1) is an exact representation of the mode s um for a flat-earth modeL An identical res ult has been give n by Wait [1960, 1962] and it also agrees with the special case [(Te=oo or Re = 1] derived by AI'pert [1955] and Budde n [1962].Howe ver, it mi ght b e me ntioned, in passing, that even for a flat earth , (1) is not complete as , in addition, there is a contribution from the "branch-line integrals." Fortunately, for the earth-ionosphere waveguid e, these are negligible , as pointed out in the quoted references.A more serious objection to (1) has to do with th e influence of e arth c urvature. Volland attempts to deal with this question by multiplyin g each ter m in the mode sum by a factor Bn which is defin e d by where 8 = p/a and a is the radius of th e earth. Clearly, the logic be hind thi s s te p is that th e factor (8/ sin 8) 112 accounts for the horizon tal convergence, while the 1465